THE UNIVERSITY OF ROCHESTER

Nuclear Science Research Group

ROCHESTER, NEW YORK 14267-0216, USA

Surface Boiling – a New Type of Instability of Highly

Excited Atomic Nuclei

J. Tõke and W. U. Schröder

UR-NCUR 10-26

Dec 2010/Jan 2011, Submitted to Physical Review C

Tri

ad

Nancy J

urs

1

Surface Boiling - a New Type of Instability of Highly Excited Atomic Nuclei

J. Toke and W.U. Schroder

Departments of Chemistry and Physics

University of Rochester, Rochester, New York 14627

ABSTRACT

The evolution of the nuclear matter density distribution with excitation

energy is studied within the framework of a finite-range interacting Fermi

gas model and microcanonical thermodynamics in Thomas-Fermi approxi-

mation. It is found that with increasing excitation energy, both infinite and

finite systems become unstable against infinitesimal matter density fluctua-

tions, albeit in different ways. In modeling, this instability reveals itself via

an apparent negative heat capacity of the system and is seen to result in

the volume boiling in the case of infinite matter and surface boiling in the

case of finite systems. The latter phenomenon of surface boiling is unique

to small systems and it appears to provide a natural explanation for the

observed saturation-like patterns in what is commonly termed caloric curves

and what represents functional dependence of nuclear temperature on the

excitation energy.

2

I. INTRODUCTION

Understanding the limits of thermodynamical stability of excited nuclear systems has

been a focus of numerous theoretical and experimental studies [1–3] from the dawn of

nuclear science. While experimental studies invariably involve finite nuclear systems

formed in the course of nuclear reactions and thus not subjected to external confine-

ment and the related pressure, theoretical studies are overwhelmingly concentrated on

instabilities in bulk nuclear matter kept under controlled conditions. This then results,

e.g., in phase separation and spinodal phenomena, the relevance of which for finite nu-

clear systems is far from obvious. The reason here is that it is not possible to bring

the bulk matter in finite nuclei to the state showing in theoretical analysis the kind of

instability of interest. Unlike in theoretical modeling, it is not possible to control the

volume, pressure, or the temperature of actual nuclei and the best one can hope for is

that by solely feeding more and more (excitation) energy into the system one will arrive

at a point where the system becomes thermodynamically unstable, in addition to being

trivially unstable against statistical particle evaporation, fragmentation, gamma, and

beta decays.

To discover such instabilities in theoretical modeling of nuclear systems, one must

then keep increasing the excitation energy, the only controlling parameter available in

experiments, in infinitesimally small steps and checking if a metastable equilibrium is

possible for a consecutive value of the excitation energy. The system is considered at

metastable thermodynamical equilibrium when it may decay only as a result of finite

statistical fluctuations in parameters describing the system as a whole. The onset of

instability is in such a case signaled by the loss by the system of immunity against

infinitesimally small fluctuations in one or more parameters, i.e., where such fluctuations

no longer would give rise to restoring driving forces, but rather to disruptive forces.

Mathematically, such instabilities reveal themselves in modeling through the appearance

of negative compressibility, negative heat capacity, or negative derivative of chemical

potential with respect to concentration [1, 4, 5].

The present work is part of a continued effort [6–16] to construct a (microcanoni-

cal) thermodynamical framework for understanding phenomena of apparently statistical

3

nature observed in highly excited nuclear systems produced in the course of heavy-ion

collisions. More specifically, it focuses on the negative heat capacity observed [10] for

a thermally expanding bulk nuclear matter considered within the interacting Fermi gas

model within Thomas-Fermi approximation. Now, for the first time, the onset of this

negative heat capacity is identified as the boiling point for bulk nuclear matter, however,

occurring at conditions substantially different from those typically attributed to boiling.

Furthermore, model calculations performed for finite systems using finite-range inter-

acting Fermi gas model combined with Thomas-Fermi approximation demonstrate that,

on the excitation energy scale, much before the bulk matter would come to boiling, the

surface matter starts boiling off, not allowing one to reach temperatures in excess of this

surface boiling-point temperature. The phenomenon of surface boiling, discovered here

via theoretical modeling, appears unique to small systems interacting via finite-range

forces, such as is the case of atomic nuclei. It finds a solid experimental confirmation in

the form of caloric curves ”saturating” at temperatures of several MeV [2], expected for

surface boiling.

The present paper is structured as follows. In Section II the adopted theoretical

formalism is presented based on finite-range interacting Fermi gas model and Thomas-

Fermi approximation in conjunction with microcanonical statistical thermodynamics.

In Section III, the boiling instability in bulk matter at zero pressure is revisited with

clear demonstration of its nature. In Section IV, the evolution of the density profile of

finite droplets of nuclear matter with increasing excitation energy is studied revealing

the onset of instability against the infinitesimal fluctuations in the local matter density

profiles, where such infinitesimal fluctuations are seen to give rise not to restoring forces

but to effective driving forces amplifying the fluctuations to the point where parts of

the surface matter separate from the system in what constitutes surface boiling. The

conclusions are presented in Section V

II. THEORETICAL FRAMEWORK

A phenomenological model has been proposed and used earlier [6–8, 10–13] to allow

treatment of excited nuclei as droplets of unconfined Fermi gas/liquid in an (approx-

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imate) microcanonical equilibrium, where all allowed microstates are assumed to be

equally probable. Since a true microcanonical equilibrium is not possible in a system

excited in excess of particle binding energy, one considers the excited nuclear system to

be bound in phase space by a hypersurface of transition states which can be reached by

way of finite fluctuations and which act as effective doorways to decay channels. Exam-

ples of such transition states are states where at least one particle is in continuum at or

in excess (for protons) of the Coulomb barrier or states at fission or fragmentation saddle

configurations. The model is valid only in the domain of excitation energies, where such

a hypersurface exists, i.e., where the system can be in a state of transient metastability,

justifying the notion of an approximate microcanonical equilibrium. The micro-states

are considered allowed when their energy is equal to the assumed model energy of in-

terest and when they obey other conservation rules. Under such assumptions, common

to all statistical models of excited nuclei, the probability for the system to reside in any

particular macrostate or configuration is proportional to the number of microstates such

configuration allows to visit, which is called here the configuration partition function

Zconfig and logarithm of which is Boltzmann’s configuration entropy, Sconfig = lnZconfig.

Note that Boltzmann’s little − k is unity according to an adopted convention, where

nuclear temperature is measured in units of MeV. The term configuration refers to a

macroscopically distinct state of the system characterized by the matter distribution

and, possibly, collective velocity field, such as, e.g., the one corresponding to collective

rotation, vibration, or hypothetical self-similar expansion [17]. In the present study,

given its goals, the collective velocity field is assumed to be absent, securing highest

entropy for any macroscopic matter distribution.

Accordingly to the notion of Boltzmann’s entropy, the probability of finding the

system in a given configuration is given by

Pconfig =Zconfig

Zsystem

=eSconfig

eSsystem, (1)

where Zsystem and Ssystem are the system partition function and entropy, respectively.

The former can be expressed as

Zsystem = ΣiZiconfig, (2)

where the sum extends over all possible configurations. Since the number of such config-

5

urations is excessively large, Eq. 2 as a whole is impractical and one is always compelled

to limit the sum on its R.H.S. to a manageably small number of configurations of interest,

expected to be dominant. While often it is sufficient to consider only the most likely con-

figuration out of the very many configurations described by a suitable parametrization of

the matter distribution, one must keep in mind that potentially interesting phenomena

are the ones where more than one configuration must be explicitly considered.

In view of the above, the important quantity to evaluate is the configuration (Boltz-

mann’s) entropy, which the present formalism approximates via the zero-temperature

interacting Fermi-gas model equation as

Sconfig = 2√

aconfig(E − Econfig), (3)

where E is the system energy and aconfig and Econfig are, respectively, the level density

parameter and zero-temperature energy for the given configuration.

Equation 3 is the base equation of the model, allowing one to evaluate Sconfig for

any configuration of interest characterized solely by the configuration matter density

distribution ρconfig(~r). Such an evaluation involves evaluating separately aconfig and

Econfig. The former is done using Thomas-Fermi approximation [6] and the latter is

done by integrating over volume the energy density given by a suitable equation of state,

with a folding provision for mocking up the effects of the finite range of nucleon-nucleon

interaction. For aconfig one writes [6]

aconfig = αoρ2/3o

∫ ∫ ∫ρ1/3(~r)d~r, (4)

where αo expresses the value of the level density parameter per nucleon at normal matter

density ρo.

The zero-temperature energy of a given configuration Econfig is taken as consisting of

a potential (interaction) energy part EEOSint and a kinetic energy part EPauli arising from

the action of the Pauli exclusion principle, i.e.,

Econfig = EEOSint + EPauli. (5)

The interaction energy is here calculated by folding a standard Skyrme-type EOS

interaction energy density εEOSint (ρ) with a Gaussian folding function with a folding length

6

λ adjusted so as to approximately reproduce the experimental surface diffuseness of finite

droplets of nuclear matter, i.e.

EEOSint = RGauss

∫ ∫ ∫ ∫ ∫ ∫εEOSint (ρ(~r − ~r′))e−

(~r−~r′)22λ2 d~rd~r′, (6)

where RGauss is the normalization factor for the folding Gaussian. Note that the use of a

folding integral of Eq. 6 makes the EOS non-local, such that the values of intensive ther-

modynamical parameters in a given location depend on the matter density distribution

in neighboring locations. This results in the same problems, albeit on a smaller scale,

that thermodynamics encounters when facing Coulomb and/or gravitational forces.

The Pauli energy was calculated from a Fermi-gas model expression

EPauli =3

5EFermi

o ρ−2/3o

∫ ∫ ∫ρ5/3(~r)d~r, (7)

where EFermio denotes the Fermi energy at normal matter density ρ = ρo, characteristic

of the EOS adopted.

For the equation of state, the present study adopted the standard form consistent with

Skyrme-type nucleon-nucleon interaction, which implies the interaction energy density

(appearing in Eq. 6) in the form

εEOSint (ρ) = ρ[a(

ρ

ρo

) +b

σ + 1(

ρ

ρo

)σ] (8)

The values of the parameters a, b and σ in Eq. 8 are determined by the requirements

for the binding energy, matter density, and the incompressibility modulus to have pre-

scribed values. The values chosen in this study of a=-62.43 MeV, b=70.75MeV, and σ

= 2.0 imply a normal density of ρo = 0.168fm−3, binding energy per nucleon at normal

density of εEOS/ρo=-16MeV, the incompressibility modulus of K = 220MeV , and Fermi

energy at normal density of EFermio =38.11MeV.

In the model calculations for uniformly distributed matter, the finite range of inter-

action is of no consequence and the configuration energy can be written simply as

Econfig(ρ) = V εEOS(ρ) (9)

where V is the system volume and εEOS(ρ) is the full configuration energy density (in-

cluding interaction and Pauli energies) given by the equation of state as a function of

matter density ρ.

7

The (microcanonical) temperature T , pressure p, and chemical potential µ were eval-

uated using the following model expressions

T = 1/(∂S

∂E∗ )V,N =

√E∗ − Econfig

aconfig

, (10)

p(ρ, T ) = T (∂S

∂V)E∗,N = ρo[

ρ2

ρo

dεEOS

dρ+

2

3αo(

ρ

ρo

)1/3T 2] , and (11)

µ(ρ, T ) = −T (∂S

∂N)V,E∗ =

ρ

ρo

[εEOS + ρodεEOS

dρ]− 1

3αo(

ρ

ρo

)−2/3T 2. (12)

In Eqs. 10-12, N represents the number of nucleons, αo represents the value of the little

a parameter per nucleon at normal density, and εEOS represents configuration energy

per nucleon.

It is perhaps amusing to note that according to Eq. 12, at T = 0 and at equilibrium

density ρ = ρo, the chemical potential µ is here equal to the average (configuration)

energy per nucleon εEOS, same as stated in the Hugenholtz-Van Hove theorem for a

much more strict treatment of nuclear interactions than ours.

With a formalism set up for evaluating entropies for configurations of interest one has

a sound thermodynamical framework for understanding a variety of statistical phenom-

ena occurring in highly excited nuclear systems. In particular, within this framework,

the decay rates into various decay channels are related to configuration entropies at the

transition-states for the decay channels of interest, i.e. states at the transition-state

hypersurface confining the model system. On the other hand, the (quasi-) equilibrium

properties of the system in its microcanonical metastable state can be inferred from find-

ing the configuration of maximum entropy among the ones deemed to be relevant. The

latter kind of modeling, pursued in the present study, includes in a natural way research-

ing the limits on metastability of excited nuclear systems, where no maximum entropy

is found within a suitably parameterized space of configurations under consideration.

Note that the above formalism makes a number of simplifying approximations, such

as neglecting the Coulomb forces, setting the effective nucleonic mass to the free nucleon

mass, neglecting iso-spin effects, using zero-temperature Fermi-gas model expressions,

etc. Also, the role of quantum effects on the Pauli energy [9] of finite systems is here

8

neglected. These approximations may be dropped when issues other than studied in this

work are to be addressed.

III. BOILING INSTABILITY IN BULK NUCLEAR MATTER

General behavior of uniform Fermi matter can be well understood from the appear-

ance of isotherms in the familiar Wan der Waals type plots. These are illustrated in

Fig. 1 for the bulk model matter with Skyrme-type EOS with a compressibility con-

stant of K=220 MeV. The isotherms are seen to feature spinodal domains of negative

compressibility, which have been often discussed in literature in the context of general

dynamical instability of the matter. The spinodal domains are generally inaccessible

to experiment, as it is not possible to bring the system as a whole to uniform density,

temperature, and pressure in these domains. Rather, the presence of these domains is

an indication for onset of boiling (higher densities) or condensation (lower densities)

instabilities as system parameters are varied, and thus of the possibility of phase separa-

tion and hypothetical phase coexistence, were it possible to transiently control pressure,

volume, and temperature. The isotherms illustrate also the presence of a critical point,

where the two (boiling and condensation) limits of the spinodal domain merge making

the two phases identical.

One must recognize, however, that while the isotherm plots in Fig. 1 are quite instruc-

tive, they are largely of academic value in the case of nuclear matter. This is so because

it is not possible to bring the nuclear matter to conditions spanned by these plots, as

there is no meaningful way to control any of the volume, pressure, and temperature

of real nuclear systems, while at the same time ensuring their bulk uniformity. What

is experimentally accessible in Fig. 1, is a mini-domain expressing the evolution of the

bulk (uniform) matter density with excitation energy. For any definite system, whether

infinite or finite, the accessible domain is strictly one-dimensional. For example, for hy-

pothetical infinitely large systems this domain degenerates into a short section of a line

(shown in Fig. 1 in dashes) at zero pressure, connecting the point A at normal density

(p=0, T=0, ρ = ρo) and ending at the boiling point B at p = 0, T = Tboil = 10.8MeV

and ρ ≈ 0.6ρo. For the bulk (excluding the non-uniform surface domain) of a finite,

9

2.0

1.5

1.0

0.5

0.0

-0.5

p (M

eV/fm

-3)

T=5MeV7

9

Tboil

13

15

17

Tcrit19

K=220MeVTboil=10.6MeV

Tcrit=17.9MeV

A B

C

D

1 2 5 10 20 50 ρο/ρ

FIG. 1: Isotherms for the model matter. The isotherm corresponding to zero-pressure boiling

temperature is shown in dotted line and the critical isotherm is shown in dash-dotted line.

The adiabatic trajectory for a hypothetical infinite system is shown in dashes (line AB), while

such for the bulk of a finite (A=100) system is shown in bold solid line (line CD).

A=100, system studied further below, the accessible domain degenerates into a short

line CD (at ρ > ρo) shown schematically in Fig. 1 in bold.

As noted above, much of the model space covered by Fig. 1 is experimentally inacces-

sible to nuclear science. Yet, the very fact that the accessible domain is so limited and

ends (with increasing excitation energy) always on a (boiling) spinodal, is indicative of

the presence of a limiting excitation energy real nuclear systems residing in vacuum, i.e.,

at zero external pressure are able to equilibrate without becoming unstable. And it is

the latter quantity that can be measured and thus compared to the model predictions.

The response of the bulk uniform nuclear matter to the excitation energy is illustrated

in Fig. 2. Here, for any given excitation energy per nucleon, the matter density was var-

ied so as to produce any of the three select pressures - the natural zero pressure, critical

pressure, and an intermediate pressure. The latter two are of academic interest only as

they require an external containment manostat. Note that, consistent with the First

Law of Thermodynamics, the requirement of zero pressure secures maximum (configu-

ration) entropy Sconfig for a given excitation energy and thus implies that the system

is microcanonical (within the model constraint of uniformity). Subsequently, (micro-

10

1.2

0.8

0.4

0.0

ρ/ρ ο

12 3 4 5 6 7 8 9

102 3 4 5 6 7 8

p=0 MeV/fm3

p=0.6 MeV/fm3

p=pcrit

EOS: K=220 MeV

a)

15

10

5

T (

MeV

)

12 3 4 5 6 7 8 9

102 3 4 5 6 7 8

b)B

F

-30

-25

-20

-15

µ (M

eV)

c)

1 2 5 10 20 50

E*/A (MeV)

FIG. 2: Evolution of the equilibrium density (top panel), temperature (middle panel), and

chemical potential (bottom panel) with excitation energy per nucleon for uniform Fermi matter

with compressibility constant of K=220 MeV. The three lines in each panel are, respectively,

for zero pressure (solid line), p=0.6 MeV/fm3 (dashes), and the critical pressure pcrit (dotted

line).

canonical) temperature T , pressure p, and chemical potential µ were determined for the

equilibrium uniform configuration using the Fermi gas model expressions of Eqs. 10, 11,

and 12.

As seen in Fig. 2, the equilibrium density decreases monotonically with increasing

excitation energy for all three values of pressure studied. On the other hand, temper-

ature reaches maximum value and the chemical potential reaches a minimum value at

E∗/A=17 MeV (zero pressure) and then they begin dropping (temperature) and increas-

ing (chemical potential) with increasing E∗. This kind of behavior is indicative of the

onset of thermodynamical instability at around E∗/A=17 MeV. In this domain, acqui-

11

sition by any small part of the system of an infinitesimally small amount of excitation

energy from neighboring parts via fluctuations would lead to a lowering of temperature

in this part, which will then cause driving even more heat to it from hotter neighbors

and a further lowering of temperature, until that part leaves the system. The latter

follows from the fact that the zero-pressure curve ends at a point F (E/AF ≈ 18MeV ),

where the entropy function no longer has a maximum as a function of the matter density

ρ. Thus, any portion of the matter that reaches point F, will expand indefinitely on its

own. Note, that it follows from the above, that the boiled off matter may be expected

to be colder than the liquid residue, a fact that should be verifiable experimentally. The

latent heat for boiling off is E/AF − E/AB ≈ 5.5MeV per nucleon.

A narration perhaps better suited for nuclear systems, where processes are mediated

by nucleon transport, is based on the behavior of the chemical potential and, specifically,

on the fact that in a spinodal domain this potential features (destabilizing) negative

first derivative with respect to concentration. In such a domain, when in one part of the

system the concentration (matter density) decreases via fluctuations, that part increases

its chemical potential and thus begins now feeding flux to neighboring parts and thus

increasing its chemical potential even further.

In fact, the system would never arrive at a configuration showing nominally negative

heat capacity or negative chemical susceptibility but would rather boil off ”offending”

parts of the matter while tending toward a metastable equilibrium. This (preequi-

librium) boiling allows the remainder to shed the excess excitation energy, leaving a

metastable residue at an excitation energy per nucleon equal to that at the boiling point

(E/AB ≈ 12.5MeV ). Fig. 2 illustrates additionally caloric curves calculated for hypo-

thetical uniform system subjected to two different non-zero external pressures. As seen

in this figure, at (hypothetical) critical pressure, the system stays always stable and

uniform, while at (hypothetical) intermediate pressure, the system would find stability

in a two-phase configuration.

According to the above analysis, the condition for the boiling can be expressed via a

set of two equations, such as

(∂S

∂ρ)E = 0 and (13)

12

FIG. 3: (Color online) Reduced two-phase configuration entropy surface for a configuration

of two equal-A subsystems with differing split of the available excitation energy E∗ between

these phases.

∂2S

∂E2= 0. (14)

Note that the presence of a domain of negative heat capacity can be inferred al-

ready from the appearance of the isotherms seen in Fig. 1. Clearly, by following the

zero-pressure line in the direction of decreasing matter density, representing the system

trajectory as a function of excitation energy, one would first cross consecutive isotherms

with increasing temperature labels, but then, after reaching the boiling-point temper-

ature, the system trajectory would cross consecutive isotherms with ever decreasing

temperature labels.

A further insight into the (volume) boiling phenomenon is provided in Fig. 3 where the

reduced two-phase configuration entropy Sconfig−Suniform surface is shown as a function

of total excitation energy and its possible division between the two equal parts of the

system. As seen in this figure, at low excitations entropy favors uniform configurations

where the two parts have equal excitation energies. With increasing excitation energy,

13

fluctuations in the excitation energy distribution grow to the point where a uniform

distribution no longer provides a fair description of the system and, accordingly, the

configuration temperature no longer provides an adequate representation of the system

temperature. With a further increase of excitation energy, the entropy favors asymmetric

split of the excitation energy between the two parts. In fact, the entropy would grow

indefinitely with an indefinite expansion of one part of the system at the expense of the

remainder until that part leaves the system or in other words, boils off. Note that in

Fig. 3 the mass numbers in the two parts of the model system are kept constant but the

volumes occupied by these parts are free to adjust so as to maximize the configuration

entropy for a given split of excitation energy.

The results show that in quantitative terms the model volume boiling temperature

of over 11 MeV is substantially higher than the limiting temperatures observed in ex-

perimental studies of caloric curves [2] making it unlikely for the volume boiling to be

responsible for the observed plateaus on these curves. On the other hand, as demon-

strated in the following section, in finite systems, the more vulnerable surface domain

begins boiling off at lower temperatures consistent with the observed plateaus on caloric

curves.

IV. SURFACE BOILING IN FINITE SYSTEMS

The model calculations for a finite system were performed for a system of 100 nucle-

ons. The folding width parameter µ was 1.4fm. The system density distribution was

parameterized in terms of error function [6] as

ρ

ρo

= C(Rhalf , d)[1− erf(r −Rhalf√

2d)], (15)

where Rhalf and d are the half-density radius and the Sussmann surface width, respec-

tively, and C(Rhalf , d) is a normalization factor assuring the desired number of nucleons

in the system (here, A=100).

The results of the calculations are displayed in six panels on Fig. 4. The various

panels illustrate the evolution of the system parameters with the excitation energy per

nucleon. As seen in this figure, the system expands (panel a) as the excitation energy

is increased up to 4 MeV per nucleon and so does the surface width (panel b) and the

14

relative surface width (panel c). Accordingly, the bulk matter density in the center of

the system decreases (panel e). The central pressure is seen to decrease (panel e) due

to the reduction in surface tension. Importantly, the caloric curve (panel f) features a

maximum followed with a domain of negative heat capacity. The negative heat capacity

is seen to set in around an excitation energy per nucleon of approximately 5 MeV. It

signals onset of an instability where a part of the surface domain may draw excitation

energy from neighboring parts and while doing so expands and cools down. As a result,

it will now draw even more energy from hotter neighbors and cool down even further

until it separates from the system in a process that can be identified as surface boiling.

The situation here is very much analogous to the case of volume boiling, except that

the surface boiling sets in at much lower temperatures consistent with weaker bonding

in the surface domain.

The loss of monotonicity at around E ∗ /A=4.5 MeV seen in panels a-e of Fig. 4 is

due to the fact that the rather arbitrarily chosen two-phase configuration entropy ceases

to provide a good approximation for the system entropy as the system approaches the

boiling point. Like in the case of infinite matter, also here the zero-pressure trajectory

of metastability ends at a point on the energy scale, where no profile secures maximum

of entropy allowing the surface diffuseness of parts of the system grow indefinitely on

their own.

V. SUMMARY

The modeling of the behavior of hot bulk matter and hot finite nuclei has revealed

that both kinds of systems are subject to boiling off of matter when brought to excita-

tion energy per nucleon exceeding certain critical (not to be confused with critical point)

value. In realistic finite systems the surface boiling sets in at considerably lower excita-

tion energies than would the volume boiling of the bulk matter, which should prevent

the system from ever experiencing volume boiling. On the other hand, the surface boil-

ing, occurring in model calculations at temperatures around 5 MeV provides a natural

explanation for the observed plateaus on caloric curves and observed limiting excitation

energies that can be equilibrated in nuclear systems. The surface boiling appears unique

15

FIG. 4: Evolution of finite system parameters with excitation energy (See text).

to small systems where the surface domain plays relatively important role. Importantly,

it appears impossible to avoid it as one “pumps” more and more excitation energy into

the system. It simply is not possible to generate excitation at a rate slow enough to

allow the excess to be carried away via statistical particle evaporation.

It appears far from obvious what form the boiled off matter may take, and specifically,

whether it will cluster occasionally into intermediate-mass fragments. However, violent

departure of parts of the surface domain may well give rise to shape fluctuation leading

to Coulomb fragmentation [13].

16

Consistent with the goals set, the present study employs a number of approximations

and simplifications, which may be dropped in follow-up studies. For example, the de-

veloped general formalism can be readily adapted for handling iso-asymmetric systems

allowing one to study isospin effects in boiling off of surface matter. Also, the quantum

effects studied in Ref. [9] should reduce the density of the bulk matter, depending on

the size of the system. One may then wish to investigate the influence of such quantum

effects on the boiling temperatures of systems of various sizes.

Acknowledgments

This work was supported by the U.S. Department of Energy grant No. DE-FG02-

88ER40414.

[1] H. Muller and B. Serot, Phys. Rev. C 52, 2072 (1995).

[2] J. Natowitz et al., Phys. Rev. C 65, 34618 (2002).

[3] S. Shlomo and V. Kolomietz, Rep¿ Prog. Phys. 68, 1 (2005).

[4] J. Lattimer and D. Ravenhall, Astr. Journ. 223, 314 (1978).

[5] M. Barranco and J. Buchler, Phys. Rev. C 22, 1729 (1980).

[6] J. Toke and W. J. Swiatecki, Nucl. Phys. A 372, 141 (1981).

[7] J. Toke and W. U. Schroder, Phys. Rev. Lett. 82, 5008 (1999).

[8] J. Toke, J. Lu, and W. U. Schroder, Phys. Rev. C 67, 034609 (2003).

[9] J. Toke and W. Schroder, Phys. Rev. C 65, 044319 (2002).

[10] J. Toke, L. Pienkowski, L. Sobotka, M. Houck, and W. U. Schroder, Phys. Rev. C 72,

031601 (2005).

[11] J. Toke, J. Lu, and W. U. Schroder, Phys. Rev. C 67, 044307 (2003).

[12] J. Toke and W. U. Schroder, Proc. IWM2005 Int. Worksh. on Multifragmentation and

Rel. Topics (Societa Italiana di Fisica, Bologna, Italy, 2006), p. 379.

[13] J. Toke and W. Schroder, Phys. Rev. C 79, 064622 (2009).

[14] L. G. Sobotka, R. J. Charity, J. Toke, and W. U. Schroder, Phys. Rev. Lett. 93, 132702

17

(2004).

[15] L. G. Sobotka, and R. J. Charity, Phys. Rev. C 73, 014609 (2006).

[16] C. Hoel, L. G. Sobotka, and R. J. Charity, Phys. Rev. C 75, 017601 (2007).

[17] W. A. Friedman, Phys. Rev. Lett. 60, 2125 (1988).

[18] N. Hugenholtz and L. Van Hove, Physica 24, 363 (1958).